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sp06e2sol

# sp06e2sol - Math 2374 Spring 2006 Midterm 2 Solution Page 1...

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Math 2374 Spring 2006 Midterm 2 Solution - Page 1 of 5 March 29, 2006 1. (25 points) Let the curve C in the ( x, y )-plane be the boundary of the unit square: C consists of four line segments, from (0 , 0) to (1 , 0); from (1 , 0) to (1 , 1); from (1 , 1) to (0 , 1); and from (0 , 1) to (0 , 0). Evaluate the line integral C xy ( - 1 + x 2 + 9) dx + 1 3 ( x 2 + 9) 3 / 2 dy by using Green’s Theorem. Solution. Need to observe first that C is a closed curve which is the boundary of the unit square D : 0 x 1, 0 y 1. Since it is closed we may use Green’s theorem: C P dx + Q dy = D Q x - P y dxdy In our case P = xy ( x 2 + 9 - 1), P y = x ( x 2 + 9 - 1) = x x 2 + 9 - x Q = 1 3 ( x 2 + 9) 3 / 2 , Q x = 1 3 3 2 2 x ( x 2 + 9) 1 / 2 = x x 2 + 9 Thus Q x - P y = x x 2 + 9 - ( x x 2 + 9 - x ) = x By Green’s theorem C P dx + Q dy = D x dxdy = 1 0 1 0 x dy dx = 1 0 x dx = 1 / 2 . Deductions. Mistake in the computation of the last integral 2 pts o ff . Wrong computation of Q x - P y leads to 5-7 pts o ff provided that integration after still done up to a ”right” number. If not, other deductions might take place.

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Math 2374 Spring 2006 Midterm 2 Solution - Page 2 of 5 March 29, 2006 2. (25 points) You build a fence so that the base of the fence is the circle C given by x 2 + y 2 = 4. If the height of the fence at any point ( x, y ) along the circle is given by f ( x, y ) = x +4, calculate the area of the fence.
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sp06e2sol - Math 2374 Spring 2006 Midterm 2 Solution Page 1...

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